Chapter A.10, Problem 66AYU

To determine

To find: Simplify the expression ${\left(16\right)}^{-\frac{3}{2}}$.

Answer to Problem 66AYU

The simplest form of ${\left(16\right)}^{-\frac{3}{2}}$ is $\frac{1}{64}$.

Explanation of Solution

Given:

It is asked to simplify the expression ${\left(16\right)}^{-\frac{3}{2}}$.

Calculation:

If $a$ is a real number and $m$ and $n$ are integers containing no common factors, with $n\ge 2$, then $a^{\frac{m}{n}}=\sqrt[n]{a^m}={\left(\sqrt[n]{a}\right)}^m$ provided that $\sqrt[n]{a}$ exists.

Consider ${\left(16\right)}^{-\frac{3}{2}}$.

${\left(16\right)}^{-\frac{3}{2}}=\frac{1}{{\left(16\right)}^{\frac{3}{2}}}=\frac{1}{\sqrt{{16}^3}}=\frac{1}{{\left(\sqrt{16}\right)}^3}=\frac{1}{4^3}=\frac{1}{64}$